Step-by-step explanation:
"integral" means here that the factors in the polynomial are integers.
we have 3 zeros.
the least degree of a polynomial with 3 zeros is 3.
and yes, that works also for 2 zeros (the degree must be at least 2, it must be at least a quadratic equation).
or 4 zeros (at least 4th degree).
it in general n zeros (at least nth degree).
constructing this out of the given zeros is easy.
what happens, when I multiply something by 0 ? the total result will be 0.
and so, we simply multiply 3 short terms with each other, where each term turns 0 for one of the given zeros.
what expression in x turns 0, when x = -1/3 ?
well : x + 1/3 or with integers 3x + 1 (multiplied by 3)
and for x = 2/3 ?
x - 2/3 or with integers 3x - 2
and for x = -1/4 ?
x + 1/4 or with integers 4x + 1
so, our polynomial function (of at least 3rd degree) is then
(3x + 1)(3x - 2)(4x + 1)
basically this could be already the result, depending on what your teacher wants.
for the fully extended form we need to do the multiplications :
(3x + 1)(3x - 2) = 9x² - 6x + 3x - 2 = 9x²- 3x - 2
(9x²- 3x - 2)(4x + 1) = 36x³ + 9x² - 12x² - 3x - 8x - 2 =
= 36x³ - 3x² - 11x - 2
the requested polynomial function is
f(x) = 36x³ - 3x² - 11x - 2
